Extendability of quadratic modules over a polynomial extension of an equicharacteristic regular local ring

We prove that a quadratic $A[T]$-module $Q$ with Witt index ($Q/TQ$)$\geq d$, where $d$ is the dimension of the equicharacteristic regular local ring $A$, is extended from $A$. This improves a theorem of the second named author who showed it when $A$ is the local ring at a smooth point of an affine variety over an infinite field. To establish our result, we need to establish a Local-Global Principle (of Quillen) for the Dickson--Siegel--Eichler--Roy (DSER) elementary orthogonal transformations.Comment: 19 page

First Record of African Angel Shark, Squatina africana (Chondricthyes: Squatinidae) in Indian Waters, Confirmed by DNA Barcoding

A single specimen of African angel shark, Squatina africana (Regan, 1908) was caught off Lakshadweep (11°5′47″ N; 72°2′21″ E), India in September 2016. The present study is a new report of the above species from Indian waters. In addition to classical methodologies, DNA barcoding was also adopted for species identification. The 650 bp-long region of mitochondrial Cytochrome Oxidase subunit I was sequenced to obtain the DNA barcode for the species under study. The sequence divergence value within species and between species was calculated using MEGA V.7.0, where Kimura 2 parameter (k2p) model was chosen as a distance model. The average k2p distance separating individuals within species was 1.76% and inter specific divergence was 8–10%. A neighbour joining network was constructed to provide a graphical representation of divergence between the species. Using the maximum identity with Gen Bank database, K2P divergence distance, NJ-network and traditional morphological approach, we could identify the given specimen as a mature male African angel shark